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Multiplying Complex Numbers: (3 - i)(3 + i)
This article will explore the process of multiplying two complex numbers, specifically (3 - i)(3 + i). We'll demonstrate the steps involved and discuss the interesting result.
Understanding Complex Numbers
Complex numbers are numbers of the form a + bi, where a and b are real numbers, and i is the imaginary unit defined as the square root of -1.
Multiplication of Complex Numbers
To multiply complex numbers, we use the distributive property, just as we do with real numbers. This means multiplying each term in the first complex number by each term in the second complex number.
Multiplying (3 - i)(3 + i)
Let's break down the multiplication:
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Expand using the distributive property: (3 - i)(3 + i) = 3(3 + i) - i(3 + i)
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Simplify: = 9 + 3i - 3i - i²
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Substitute i² = -1: = 9 + 3i - 3i - (-1)
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Combine like terms: = 9 + 1
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Final result: = 10
The Interesting Result
The product of (3 - i) and (3 + i) is 10, a real number! This is a special case of a more general principle: the product of a complex number and its conjugate (which is the same number but with the sign of the imaginary part reversed) always results in a real number.
Conclusion
We've seen how to multiply complex numbers and discovered a special property: the product of a complex number and its conjugate is always real. This example highlights the unique nature of complex numbers and their interesting properties.